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We report semiconductor to metal-like crossover in temperature dependence ofresistivity ($\rho$) due to the switching of charge transport from bulk tosurface channel in three-dimensional topological insulatorBi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$. Unlike earlier studies, a muchsharper drop in $\rho$($T$) is observed below the crossover temperature due tothe dominant surface conduction. Remarkably, the resistivity of the conductingsurface channel follows a rarely observable $T^2$ dependence at low temperatureas predicted theoretically for a two-dimensional Fermi liquid system. The fielddependence of magnetization shows a cusp-like paramagnetic peak in thesusceptibility ($\chi$) at zero field over the diamagnetic background. The peakis found to be robust against temperature and decays linearly with field fromits zero-field value. This unique behavior of $\chi$ is associated with thespin-momentum locked topological surface state ofBi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$. The reconstruction of surface statewith time is clearly reflected through the reduction of peak height with theage of the sample.

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Prominent metallic surface conduction and the singular magnetic response  of topological Dirac fermion in three-dimensional topological insulator  Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$